In the following lecture, starting at minute 29:00 and going further, the professor resolves the Twin Paradox using Lorentz velocity addition. I have a question about this:

Isn't the figure given below (taken from a slide in the lecture, at time 37:50) referring to the case where there is an independent object moving relative to the inertial frames S and S'? In the example of the twins that he discusses, there is no object independent of the frames of reference. We simply have the twins, each attached to her respective reference frame. Why, then, does he use the formula for Lorentz velocity addition? What object is he referring to?

1 Answer
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Briefly skimming the video, it seems like he's trying to resolve the paradox by having the twins meet up again somewhere else than their initial position (see 35:55). Basically, the traveling twin keeps going, and then the stationary twin tries to overtake her, starting at a later time. The velocity addition rule is used to find how fast the lingering twin needs to go to catch up so they both meet at a certain point.

The twin paradox is what happens when people don't realize there are three reference frames in play - the stationary one, the outgoing one, and the ingoing one. By treating the latter two as the same, one gets nonsensical results. The usual resolution comes from carefully drawing a couple spacetime diagrams and making sure whatever quantity you're considering is well defined.

This video tries to eliminate the problem in a way I've never seen before. It may be novel, but I've also never seen the twin paradox require so much cumbersome arithmetic. Also, there are still three different frames in this case - the stationary one, the outgoing one, and the even faster outgoing one. I'm doubtful whatever resolution comes from this will result in a better intuition for SR.